In this work, we present a high-order finite volume framework for the numerical simulation of shallow water flows. The method is designed to accurately capture complex dynamics inherent in shallow water systems, particularly suited for applications such as tsunami simulations. The arbitrarily high-order framework ensures precise representation of flow behaviors, crucial for simulating phenomena characterized by rapid changes and fine-scale features. Thanks to an {\it ad-hoc} reformulation in terms of production-destruction terms, the time integration ensures positivity preservation without any time-step restrictions, a vital attribute for physical consistency, especially in scenarios where negative water depth reconstructions could lead to unrealistic results. In order to introduce the preservation of general steady equilibria dictated by the underlying balance law, the high-order reconstruction and numerical flux are blended in a convex fashion with a well-balanced approximation, which is able to provide exact preservation of both static and moving equilibria. Through numerical experiments, we demonstrate the effectiveness and robustness of the proposed approach in capturing the intricate dynamics of shallow water flows, while preserving key physical properties essential for flood simulations.

翻译：本文提出了一种用于浅水流动数值模拟的高阶有限体积框架。该方法旨在精确捕捉浅水系统中固有的复杂动力学特性，特别适用于海啸模拟等应用场景。该任意高阶框架确保了流动行为的精确表示，对于模拟以快速变化和精细尺度特征为特征的现象至关重要。通过基于生产-破坏项的特制重构，时间积分在不施加任何时间步长限制的情况下保证了保正性，这对物理一致性至关重要——尤其是在负水深重构可能导致非物理结果的场景中。为引入由底层平衡律支配的通用稳态平衡保持能力，高阶重构与数值通量以凸方式与平衡近似混合，该近似能够同时精确保持静态平衡和移动平衡。通过数值实验，我们展示了所提方法在捕捉浅水流动复杂动力学特性方面的有效性和鲁棒性，同时保留了洪水模拟所需的关键物理属性。